1. The problem statement, all variables and given/known data 2. The shape of a hill is described by the height function h(x,y) = (2 + x2 + y4)-1/2 (a) Find the gradient ∇h(x,y) (b) What is the maximum slope of the hill at the point r0 = i+j [or (x,y) = (1,1)]? (c) If you walk north-east (in the direction of the vector i+j) from the point r0, what is the slope of your path? Are you going uphill or downhill? (d) Find a vector in the direction of a path through the point r0 that remains at the same height. 2. Relevant equations 3. The attempt at a solution For parts a), b) and c) I think I'm okay. a) ∇h(x,y) = ([-x(2 + x2 + y4)-3/2], [-y3(2 + x2 + y4)-3/2]) b) At point (1,1) just plug x and y into ∇h(x,y) to get (-1/8 , -1/4) c) Along path of vector (i+j) you will be going in a downhill direction since it is in a direction 'away' from the maximum slope found in the previous question. But is there a way I can mathematically prove this? d) This is the part I'm unsure of. It's been a while since I did work with vectors. I'm guessing I need to find a vector such that infinitesimal change in the x-y direction gives no change in height. So that would mean d(h(x,y)) = 0...? Since this would signify no change in the value of h(x,y). Overall I'm a bit lost. Thanks.